3. Taking a walk in the Garden of Knowledge Speaker: Ambjörn Naeve Affiliation: Centre for user-oriented IT-Design (CID) Dept. of Numerical Analysis and Computing Science Royal Institute of Technology (KTH) Stockholm, Sweden email: amb@nada.kth.se web-sites: cid.nada.kth.se kmr.nada.kth.se

4. What follows are some snapshots from a walk in the third prototype of the Garden of Knowledge. The prototype is available on CD-rom and part of it is accessible on the web at the address: http://cid.nada.kth.se/il/kt/ktproto This prototype was developed at CID during 1996/97 in collaboration with the Royal College of Music and the Shift New Media Group.

5. The entrance to the Garden of Knowledge

6. The overall subject patch Clicking ”geometri” opens the geometry patch.

7. Overall view of the geometry patch

8. Browsing the geometry patch Clicking the left margin returns to the overall subject patch.

9. Pointing to ”symmetri” produces a definition Symmetry = invariance under motion.

10. Clicking ”symmetri” opens a new sublevel Pointing to ”rosetter” shows preview of content.

11. Pointing to ”band” shows preview of content

12. Clicking ”band” opens a new sublevel

13. Clicking ”make your own bands” brings up a tool which lets you create bands according to the 7 different symmetry types.

14. Clicking on ”exercises” lets you practice and guess which symmetry elements that are present in a chosen band. Is there a horizontal reflection? Is there a vertical reflection? Is there a half turn? Is there a glide reflection? choose a band Exercises

15. Clicking ”länkar” leads to musical symmetries (länkar = links)

16. Clicking the notes plays the pitch symmetry (without rhythm symmetry elements) pitch symmetry

17. Clicking ”rytmsymmetri” adds symmetries of rythm (with the possible choices of translation or reflection)

18. Clicking ”fördjupning” gives deeper explanation of why there are 7 different types of band symmetries. Clicking “band” prompts you to try to generate the two missing symmetries by choosing different combinations of L, V, G, H. Active area Click the band to activate / clean it.

19. Here we have found the 7:th symmetry type

20. In ”regler” we check all combinations of L,V,G,H showing which combinations that give valid symmetry types. rules: rules

21. And which combinations that break the rules rules:

22. In “bevis” we prove that L,V,G,H generate the symmetries Step1: Showing that a plane isometry is determined by how it maps 3 points. Step2: Showing that such a mapping can be achieved by ≤ 3 reflections in lines. proof

23. Step 3: proving a lemma Lemma: Reflections in two lines L 1 and L 2 is invariant under rotation of the lines around their point of intersection.

24. Step 4: applying the lemma to reflection in 3 lines First: Rotate the two first lines so that the second is perpendicular to the third. Second: Rotate the two last lines so that the second is parallel to the first. Conclusion: Reflection in three arbitrary positioned lines is a glide reflection.

25. Clicking “glidspegling” shows a glide reflection

26. Clicking “tapeter” opens a new sublevel (tapeter = wallpapers)

27. Clicking “make your own wallpapers” brings up a tool which lets you experiment with the 17 different wallpaper symmetries.

28. Clicking “Alhambra” takes you to a web-site which contains examples of all the 17 symmetry types in islamic art.

29. One of the patterns displayed at Alhambra

30. Another one of the patterns at Alhambra